BookyAI asked "Move BookyAI to Applications?" when it opened. Is it safe to say yes?

Yes. From version 1.9.19, on a Mac, BookyAI checks at start whether it is running from your Applications folder. If it is not (for example from Downloads), it offers to move itself there, because outside Applications macOS keeps an app in a protected copy that updates cannot replace. "Move to Applications" copies the app into Applications and reopens it from there; if an older copy is already in Applications it is replaced. Your books, settings and keys are stored elsewhere and are not touched. "Not now" leaves everything as it is; tick "Don't ask again" and BookyAI stops asking. You can do the same later from the update panel (Settings, Extra tools, Updates), or by dragging the app into Applications in Finder.